Prof in vector space of continuity of Fouriercoefficients of complex series

Help need.

Consider the vector space of limited complex series of the uniform norm.

Prove that the Fouriercoefficient c(f) given by the inner product space (from -phi to phi) of the complex conjugate of two functions (f(x)*g(x), where f(x) just is a function and that g(x) in the conjugate), defines a norm reducing function and conclude from this that this function is continuers.

The n-th Fourier coefficient is (if I remember correctly) exactly the inner product of f and a function f_n from a fixed orthonormal base.
So you have:
c_n=<f,f_n> (where <.,.> denotes the inner product)
and by Cauchy-Schwarz you have
|c_n|<=||f||.||f_n||=||f||.

But maybe I misunderstood your question and you want to put all coefficients in the result of the function. If yes, which sequence space are you working in? l_\infty?